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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.ellint.ellint_3"></a><a class="link" href="ellint_3.html" title="Elliptic Integrals of the Third Kind - Legendre Form">Elliptic Integrals of the
      Third Kind - Legendre Form</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.ellint.ellint_3.h0"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_3.synopsis"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.synopsis">Synopsis</a>
      </h5>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">ellint_3</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">phi</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<h5>
<a name="math_toolkit.ellint.ellint_3.h1"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_3.description"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.description">Description</a>
      </h5>
<p>
        These two functions evaluate the incomplete elliptic integral of the third
        kind <span class="emphasis"><em>Π(n, φ, k)</em></span> and its complete counterpart <span class="emphasis"><em>Π(n,
        k) = E(n, π/2, k)</em></span>.
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/ellint_3.svg" align="middle"></span>

        </p></blockquote></div>
<p>
        The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
        type calculation rules</em></span></a> when the arguments are of different
        types: when they are the same type then the result is the same type as the
        arguments.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">phi</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
        Returns the incomplete elliptic integral of the third kind <span class="emphasis"><em>Π(n,
        φ, k)</em></span>:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint4.svg"></span>

        </p></blockquote></div>
<p>
        Requires <span class="emphasis"><em>k<sup>2</sup>sin<sup>2</sup>(phi) &lt; 1</em></span> and <span class="emphasis"><em>n &lt; 1/sin<sup>2</sup>(φ)</em></span>,
        otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        (outside this range the result would be complex).
      </p>
<p>
        The final <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
        be used to control the behaviour of the function: how it handles errors,
        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">policy
        documentation for more details</a>.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
        Returns the complete elliptic integral of the first kind <span class="emphasis"><em>Π(n, k)</em></span>:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint8.svg"></span>

        </p></blockquote></div>
<p>
        Requires <span class="emphasis"><em>|k| &lt; 1</em></span> and <span class="emphasis"><em>n &lt; 1</em></span>,
        otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        (outside this range the result would be complex).
      </p>
<p>
        The final <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
        be used to control the behaviour of the function: how it handles errors,
        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">policy
        documentation for more details</a>.
      </p>
<h5>
<a name="math_toolkit.ellint.ellint_3.h2"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_3.accuracy"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.accuracy">Accuracy</a>
      </h5>
<p>
        These functions are computed using only basic arithmetic operations, so there
        isn't much variation in accuracy over differing platforms. Note that only
        results for the widest floating point type on the system are given as narrower
        types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
        zero error</a>. All values are relative errors in units of epsilon.
      </p>
<div class="table">
<a name="math_toolkit.ellint.ellint_3.table_ellint_3"></a><p class="title"><b>Table 8.65. Error rates for ellint_3</b></p>
<div class="table-contents"><table class="table" summary="Error rates for ellint_3">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> long double
                </p>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> double
                </p>
              </th>
<th>
                <p>
                  Sun compiler version 0x5150<br> Sun Solaris<br> long double
                </p>
              </th>
<th>
                <p>
                  Microsoft Visual C++ version 14.1<br> Win32<br> double
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Elliptic Integral PI: Mathworld Data
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 475ε (Mean = 86.3ε)</span><br> <br>
                  (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> <span class="red">Max = +INFε (Mean
                  = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_ellint_3__cmath__Elliptic_Integral_PI_Mathworld_Data">And
                  other failures.</a>)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
                  2.1:</em></span> Max = 1.48e+05ε (Mean = 2.54e+04ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_ellint_3_GSL_2_1_Elliptic_Integral_PI_Mathworld_Data">And
                  other failures.</a>)
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 475ε (Mean = 86.3ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 565ε (Mean = 102ε)</span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Elliptic Integral PI: Random Data
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 4.54ε (Mean = 0.895ε)</span><br> <br>
                  (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> <span class="red">Max = 3.37e+20ε (Mean
                  = 3.47e+19ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_ellint_3__cmath__Elliptic_Integral_PI_Random_Data">And
                  other failures.</a>)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
                  2.1:</em></span> Max = 633ε (Mean = 50.1ε))
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 4.49ε (Mean = 0.885ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 8.33ε (Mean = 0.971ε)</span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Elliptic Integral PI: Large Random Data
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 3.7ε (Mean = 0.893ε)</span><br> <br>
                  (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> <span class="red">Max = 2.52e+18ε (Mean
                  = 4.83e+17ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_ellint_3__cmath__Elliptic_Integral_PI_Large_Random_Data">And
                  other failures.</a>)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.557ε (Mean = 0.0389ε)</span><br>
                  <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 40.1ε (Mean = 7.77ε))
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 3.7ε (Mean = 0.892ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 2.86ε (Mean = 0.944ε)</span>
                </p>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><h5>
<a name="math_toolkit.ellint.ellint_3.h3"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_3.testing"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.testing">Testing</a>
      </h5>
<p>
        The tests use a mixture of spot test values calculated using the online calculator
        at <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>,
        and random test data generated using NTL::RR at 1000-bit precision and this
        implementation.
      </p>
<h5>
<a name="math_toolkit.ellint.ellint_3.h4"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_3.implementation"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.implementation">Implementation</a>
      </h5>
<p>
        The implementation for Π(n, φ, k) first siphons off the special cases:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="serif_italic"><span class="emphasis"><em>Π(0, φ, k) = F(φ, k)</em></span></span>
        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="serif_italic"><span class="emphasis"><em>Π(n, π/2, k) = Π(n, k)</em></span></span>
        </p></blockquote></div>
<p>
        and
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint23.svg"></span>

        </p></blockquote></div>
<p>
        Then if n &lt; 0 the relations (A&amp;S 17.7.15/16):
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint24.svg"></span>

        </p></blockquote></div>
<p>
        are used to shift <span class="emphasis"><em>n</em></span> to the range [0, 1].
      </p>
<p>
        Then the relations:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="serif_italic"><span class="emphasis"><em>Π(n, -φ, k) = -Π(n, φ, k)</em></span></span>
        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="serif_italic"><span class="emphasis"><em>Π(n, φ+mπ, k) = Π(n, φ, k) + 2mΠ(n, k)
          ; n &lt;= 1</em></span></span>
        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="serif_italic"><span class="emphasis"><em>Π(n, φ+mπ, k) = Π(n, φ, k) ; n &gt; 1</em></span>
                  
<a href="#ftn.math_toolkit.ellint.ellint_3.f0" class="footnote" name="math_toolkit.ellint.ellint_3.f0"><sup class="footnote">[4]</sup></a></span>
        </p></blockquote></div>
<p>
        are used to move φ to the range [0, π/2].
      </p>
<p>
        The functions are then implemented in terms of Carlson's integrals using
        the relations:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint25.svg"></span>

        </p></blockquote></div>
<p>
        and
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint26.svg"></span>

        </p></blockquote></div>
<div class="footnotes">
<br><hr style="width:100; text-align:left;margin-left: 0">
<div id="ftn.math_toolkit.ellint.ellint_3.f0" class="footnote"><p><a href="#math_toolkit.ellint.ellint_3.f0" class="para"><sup class="para">[4] </sup></a>
            I haven't been able to find a literature reference for this relation,
            but it appears to be the convention used by Mathematica. Intuitively
            the first <span class="emphasis"><em>2 * m * Π(n, k)</em></span> terms cancel out as the
            derivative alternates between +∞ and -∞.
          </p></div>
</div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
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